"Aw, damn it, I lost my train of thought!"
Doing math problems was like writing a novel: when inspiration flowed, you could write ten chapters in a day; when it dried up, you might manage only one chapter in ten days.
Just as Shen Qi was about to crack the key to the first geometry problem, the Vice President interrupted his train of thought.
"This uncle is seriously annoying." Shen Qi had to sort through his reasoning all over again, which cost him an extra five minutes.
His proof filled the entire sheet of paper, and at last Shen Qi found the value of sin ψ.
The answer surprised him: sin ψ was actually 1/2, which meant the angle was 30 degrees.
He measured it with a ruler. It looked like 30 degrees.
He checked it with Lobachevsky's graphing method. Sure enough, it was 30 degrees.
"I'm an idiot, a real idiot..." Shen Qi realized he'd made a basic mistake. He'd been misled by the complicated geometric diagram, and spent nearly an hour deriving the answer directly.
If he'd first used Lobachevsky's graphing method to calculate the degree of ψ, then worked backward to verify that ψ was 30 degrees, he could have saved at least half the time.
Of course, he couldn't draw with Lobachevsky's graphing method on the exam paper, but there was no problem with using it on scratch paper.
It was easier to verify a conclusion once you had one than to derive an unknown.
This was a game of wits, a mathematical contest between the person setting the questions and the person answering them.
War was the best training, and exams were the best review.
Although the system was helping him, Shen Qi had gained a new understanding of mathematics during this extremely difficult semifinal.
There had to be a deeper relationship between assumptions and proofs. Truth and fallacy weren't as simple as they seemed on the surface; a fallacy might even be the reverse derivation of a truth.
Shen Qi's mind wandered. He thought about all sorts of things for several minutes. His math level was still too low—there were so many problems he couldn't figure out.
"Two hours remain. Please hurry up and answer the questions," a proctor told all the contestants.
"Two hours, and two problems left." Shen Qi came back to himself and moved on to the next problem.
This semifinal paper had fewer questions than any he'd ever done: only three.
It was also the most difficult. The first question, worth the fewest points, had taken Shen Qi an hour.
The second question was an algebra problem. It went like this:
1
1-1
1-2-1
1-3-3-1
1-4-6-4-1
1-5-10-10-5-1
1. Calculate the sum of all the numbers in the 1024th row. (5 points)
2. Prove that the result applies to any number in the 4201st row, whether it is a fraction or a negative number. (15 points)
Quite a few high school students recognized this numerical triangle. Who didn't know Yang Hui's Triangle? Any middle school student who had taken part in a math league or an Olympiad knew its pattern.
Shen Qi was, of course, familiar with the triangle. In China, it was called Yang Hui's Triangle; in the West, it was called Pascal's Triangle, named after two mathematicians from the East and West.
The pattern was easy to spot: every number in the triangle was the sum of the two numbers immediately above it.
Following that pattern, Shen Qi quickly calculated that the sum of all the numbers in the 1024th row was xxxxx... an astronomical number, expressed as 2 to the power of 1023.
The first part of the second question was practically a freebie, which was why it was worth only 5 points.
The second part was the difficult one, worth 15 points.
Proving directly that the result applied to any number in the 4201st row, whether it was a fraction or a negative number, was a real headache. He had no idea where to start and couldn't find the slightest clue.
Shen Qi wanted to work backward. The statement he had to prove in the second part had to be based on some formula, theorem, or corollary.
"Bernoulli's permutations and combinations, or Probability Theory? No, that doesn't seem right."
"Vieta's expansion for the three special types of equations? That's not it either."
"Fermat was a master of pure number games. That's it, it has to be Fermat. He and Pascal were close friends and often corresponded, and this problem is based on Pascal's Triangle."
"Fermat came up with hundreds of conjectures in his life, and later mathematicians proved 99% of them. He was called the 'Prince of Amateur Mathematicians,' but I refuse to believe his mathematical ability was as amateurish as mine."
"My head's going to explode. Fermat had 273 conjectures, and I've studied at most 70 of them. Which one is it? Is this beyond the limits of my mathematical knowledge?"
Shen Qi set down his ballpoint pen, closed his eyes, and racked his brains for an answer.
The Vice President wandered over and stopped behind Shen Qi again. A pleased smile appeared on his face as he thought, Young man, so what if you can solve the first geometry problem and know the pattern in Yang Hui's Triangle? What difference does it make? The second part of my question is beyond your imagination, isn't it?
Just then, Shen Qi suddenly opened his eyes, which shone brightly. "I've got it! It's Fermat's (1+a) corollary! He said the corollary was definitely valid and didn't need proof, but a hundred years after his death, one of his French countrymen proved it."
The idea had come! It had come!
Inspiration surged through Shen Qi like a spring. He picked up his ballpoint pen, ready to solve the problem.
Just as he was about to start writing, Shen Qi felt a chill coming from behind him, exactly like before.
Shen Qi glanced over his shoulder, thoroughly annoyed. "Damn it, you again! Just passing by again?"
The Vice President feigned composure. "The exam hall isn't that big. As a proctor, I do happen to pass by now and then." With that, he clasped his hands behind his back and walked away.
"Uncle, you're seriously annoying!" Shen Qi calmed himself as quickly as he could and began proving the second part of the second question.
Ten minutes later, Shen Qi had written out the entire proof for the second part. Verified using Fermat's (1+a) corollary, the proof was completely valid and applied to any number in the 4201st row, whether it was a fraction or a negative number.
"I've cracked the first and second questions. That's 35 points in the bag."
Having solved two questions in a row, Shen Qi was brimming with confidence. He started on the last one.
The Vice President couldn't sit still for long. After a lap around the room, he circled behind the other contestants to watch them in secret.
"He Tao, this kid's no good. Apart from knowing Yang Hui's Triangle, he can't even begin the other problems."
"Luo Ming, from Nangang City Foreign Language School. He's from a prestigious school, after all. This kid's not bad. He got the first geometry problem right, but he's clearly botched the second part of the second question. Young man, you're going the wrong way—your whole approach is off."
The Vice President judged that Shen Qi was the fastest and most accurate contestant in the entire exam hall. He just couldn't help himself—he really couldn't—so he crept behind Shen Qi again to watch him in secret.
Shen Qi suddenly turned around, on the verge of a breakdown. "Uncle, could you not do that? Why do you have to keep picking on me?"
The Vice President looked innocent. "I set this exam paper. I'm just taking a look. Can't I do that without saying anything?"
"Damn, you set this paper?" Shen Qi couldn't believe that the expert he'd imagined setting the questions was actually this sleazy uncle.
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